2. Ordinary_Differential_Equations

2. Ordinary_Differential_Equations - Ordinary Differential...

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Unformatted text preview: Ordinary Differential Equations (ODEs) O.D.E.s are relevant to various engineering applications and will be appear in a number of 3 rd and 4 th year courses. This handout provides a summary of some common types of ODEs covered in ME 203. Note that here y represents any function of one independent variable x , where x can be any independent variable. O.D.E. General Solution How the Solution is Found = + kx dx dy kx Ce y = separate and integrate or characteristic equation (below) ) ( ) ( x g y x p dx dy = + 1 st order, linear ODE [ ] + = C dx x g y ) ( 1 where = dx x p e y ) ( 1 st order linear ODE formula was developed from integrating factors 2 2 2 = + y a dx y d ) sin( ) cos( 2 1 ax C ax C y + = characteristic equation 2 2 2 = y a dx y d ax ax e C e C y + = 2 1 or ) sinh( ) cosh( 2 1 ax A ax A y + = characteristic equation 2 2 = + + cy dx dy b dx y d a Depends on the roots of characteristic equation (i) real, unequal roots: x r x r e C e C y 2 1 2 1 + = (ii)...
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This note was uploaded on 09/16/2011 for the course ME 303 taught by Professor Serhiyyarusevych during the Spring '10 term at Waterloo.

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